| L(s) = 1 | + 3·4-s + 4·5-s − 16·7-s + 7·9-s − 20·11-s + 9·16-s + 8·17-s + 12·20-s + 6·23-s + 16·25-s − 48·28-s − 64·35-s + 21·36-s − 6·43-s − 60·44-s + 28·45-s + 28·47-s + 108·49-s − 80·55-s + 20·61-s − 112·63-s + 30·64-s + 24·68-s − 36·73-s + 320·77-s + 36·80-s + 29·81-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.78·5-s − 6.04·7-s + 7/3·9-s − 6.03·11-s + 9/4·16-s + 1.94·17-s + 2.68·20-s + 1.25·23-s + 16/5·25-s − 9.07·28-s − 10.8·35-s + 7/2·36-s − 0.914·43-s − 9.04·44-s + 4.17·45-s + 4.08·47-s + 15.4·49-s − 10.7·55-s + 2.56·61-s − 14.1·63-s + 15/4·64-s + 2.91·68-s − 4.21·73-s + 36.4·77-s + 4.02·80-s + 29/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.471195965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471195965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 - 3 T^{2} - 3 T^{6} + 29 T^{8} - 3 p^{2} T^{10} - 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 - 7 T^{2} + 20 T^{4} - 77 T^{6} + 319 T^{8} - 77 p^{2} T^{10} + 20 p^{4} T^{12} - 7 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 8 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 42 T^{2} + 1005 T^{4} - 17682 T^{6} + 249404 T^{8} - 17682 p^{2} T^{10} + 1005 p^{4} T^{12} - 42 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 4 T - 2 T^{2} + 64 T^{3} - 237 T^{4} + 64 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 3 T - 38 T^{2} - 3 T^{3} + 1473 T^{4} - 3 p T^{5} - 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 91 T^{2} + 4560 T^{4} - 185549 T^{6} + 6174239 T^{8} - 185549 p^{2} T^{10} + 4560 p^{4} T^{12} - 91 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 119 T^{2} + 5461 T^{4} + 119 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 63 T^{2} + 3729 T^{4} + 63 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 74 T^{2} + 2365 T^{4} + 18574 T^{6} - 2352596 T^{8} + 18574 p^{2} T^{10} + 2365 p^{4} T^{12} - 74 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 3 T - 18 T^{2} - 177 T^{3} - 1507 T^{4} - 177 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 14 T + 73 T^{2} - 406 T^{3} + 3708 T^{4} - 406 p T^{5} + 73 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 187 T^{2} + 20760 T^{4} - 1606517 T^{6} + 95777279 T^{8} - 1606517 p^{2} T^{10} + 20760 p^{4} T^{12} - 187 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 151 T^{2} + 11340 T^{4} - 679349 T^{6} + 39866879 T^{8} - 679349 p^{2} T^{10} + 11340 p^{4} T^{12} - 151 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 10 T - 27 T^{2} - 50 T^{3} + 6308 T^{4} - 50 p T^{5} - 27 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 98 T^{2} + 5445 T^{4} + 472262 T^{6} - 45966196 T^{8} + 472262 p^{2} T^{10} + 5445 p^{4} T^{12} - 98 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 34 T^{2} - 8715 T^{4} + 7174 T^{6} + 66197564 T^{8} + 7174 p^{2} T^{10} - 8715 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( 1 - 236 T^{2} + 29610 T^{4} - 3210544 T^{6} + 293930579 T^{8} - 3210544 p^{2} T^{10} + 29610 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( 1 + 14 T^{2} + 1125 T^{4} - 234794 T^{6} - 63431716 T^{8} - 234794 p^{2} T^{10} + 1125 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 123 T^{2} + 5280 T^{4} + 1103187 T^{6} - 133875241 T^{8} + 1103187 p^{2} T^{10} + 5280 p^{4} T^{12} - 123 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.29718762701582271713239906709, −4.97159364714456311540105447306, −4.92491777165995492427319017196, −4.80730815582992087851750190704, −4.43952244000968802725628659088, −4.40859455141446601374037759960, −4.31088120167105118063159066102, −3.82405056722635267195894701418, −3.60829543867379973801255579449, −3.48238461596148140277726126414, −3.48206579443765705149141181170, −3.38304323556668170073207205356, −3.05045660374596977411026593366, −2.94640953031324208487445812747, −2.89159646080969149310415013250, −2.66277991994484662814016728473, −2.64716463194472362082089146541, −2.51458449203497337196851573509, −2.40777570827185140436284319101, −1.94103105482411110932636228244, −1.87537496373010204920326085551, −1.36524473342694005365934633458, −1.05486642097062678618726762533, −0.69787941904193311913717862819, −0.31816310817310964324774657933,
0.31816310817310964324774657933, 0.69787941904193311913717862819, 1.05486642097062678618726762533, 1.36524473342694005365934633458, 1.87537496373010204920326085551, 1.94103105482411110932636228244, 2.40777570827185140436284319101, 2.51458449203497337196851573509, 2.64716463194472362082089146541, 2.66277991994484662814016728473, 2.89159646080969149310415013250, 2.94640953031324208487445812747, 3.05045660374596977411026593366, 3.38304323556668170073207205356, 3.48206579443765705149141181170, 3.48238461596148140277726126414, 3.60829543867379973801255579449, 3.82405056722635267195894701418, 4.31088120167105118063159066102, 4.40859455141446601374037759960, 4.43952244000968802725628659088, 4.80730815582992087851750190704, 4.92491777165995492427319017196, 4.97159364714456311540105447306, 5.29718762701582271713239906709
Plot not available for L-functions of degree greater than 10.
